Question

Given the system of inequalties:

4x-y≤3

-3x-2y≥4

Select all solutions

a
(-2,0)

b
(0,0)

c
(-5,3)

d
(5,0)

e
(-5,-5)

Answer 1

To determine which of the provided points are solutions to the system of inequalities, we need to test each point against both inequalities:

1. **4x - y ≤ 3**
2. **-3x - 2y ≥ 4**

Let's test each point:

### a. (-2, 0)

1. **For 4x - y ≤ 3**
\[
4(-2) - 0 = -8 \leq 3 \quad \text{(True)}
\]
2. **For -3x - 2y ≥ 4**
\[
-3(-2) - 2(0) = 6 \geq 4 \quad \text{(True)}
\]

**(-2, 0) is a solution.**

### b. (0, 0)

1. **For 4x - y ≤ 3**
\[
4(0) - 0 = 0 \leq 3 \quad \text{(True)}
\]
2. **For -3x - 2y ≥ 4**
\[
-3(0) - 2(0) = 0 \geq 4 \quad \text{(False)}
\]

**(0, 0) is NOT a solution.**

### c. (-5, 3)

1. **For 4x - y ≤ 3**
\[
4(-5) - 3 = -20 - 3 = -23 \leq 3 \quad \text{(True)}
\]
2. **For -3x - 2y ≥ 4**
\[
-3(-5) - 2(3) = 15 - 6 = 9 \geq 4 \quad \text{(True)}
\]

**(-5, 3) is a solution.**

### d. (5, 0)

1. **For 4x - y ≤ 3**
\[
4(5) - 0 = 20 \leq 3 \quad \text{(False)}
\]
2. **For -3x - 2y ≥ 4**
\[
-3(5) - 2(0) = -15 \geq 4 \quad \text{(False)}
\]

**(5, 0) is NOT a solution.**

### e. (-5, -5)

1. **For 4x - y ≤ 3**
\[
4(-5) - (-5) = -20 + 5 = -15 \leq 3 \quad \text{(True)}
\]
2. **For -3x - 2y ≥ 4**
\[
-3(-5) - 2(-5) = 15 + 10 = 25 \geq 4 \quad \text{(True)}
\]

**(-5, -5) is a solution.**

### Summary:
The solutions are points:
- **(-2, 0)**
- **(-5, 3)**
- **(-5, -5)**

Thus, the selected solutions are: **a, c, and e.**